Inception Neural Networks for Complete Intersection Calabi-Yau Manifolds

XVI Avogadro Meeting - December 22nd, 2020

String Theory Compactification

Superstrings defined in \(D = 10 \quad \Rightarrow \quad \mathcal{M}^{1, 9} = \mathcal{M}^{1, 3} \otimes X_6\)

Requirements

\(X_6\) is a compact manifold \((M, g)\)
\(N = 1\) SUSY in 4D
SM \(\subset\) arising gauge algebra

Solution

\(\dim_{\mathbb{C}} M = 3\)
\(\mathrm{Hol}(g) \subseteq \mathrm{SU}(3)\)
\(\mathrm{Ric}(g) \equiv 0\) or \(c_1(M) \equiv 0\)

Calabi-Yau Manifolds

no known metric for compact CY
need to study topology (Hodge numbers) to infer 4D properties

\[h^{r,s} = \dim_{\mathbb{C}} \mathrm{H}_{\overline{\partial}}^{r,s}(M, \mathbb{C})\]

Complete Intersection Calabi-Yau Manifolds

Systems of \(k\) homogeneous equations from products of \(m\) projective spaces

\[ \sum\limits_{r = 1}^m p_{\alpha}^{i_r}\, \left( z_{i_r} \right)^{a^r_{\alpha}} = 0 \quad \rightarrow \quad X = \begin{bmatrix} \mathbb{P}^{n_1} & | & a^1_1 & \cdots & a^1_k \\ \vdots & & \vdots & \ddots & \vdots \\ \mathbb{P}^{n_m} & | & a^m_1 & \cdots & a^m_k \end{bmatrix} \]

such that

\[ \begin{matrix} \text{degree of eq.} & \text{dim. of CY} & c_1 \equiv 0 \\ \Downarrow & \Downarrow & \Downarrow \\ a^r_{\alpha} \in \mathbb{N} & \dim_{\mathbb{C}} X = \sum\limits_{r = 1}^m n_r - k = 3 & n_r + 1 = \sum\limits_{\alpha = 1}^k a^r_{\alpha} \end{matrix} \]

where \(a^r_{\alpha}\) are powers of coordinates on \(\mathbb{P}^{n_r}\) in equation \(\alpha\).

Available Data

\(\exists\) compiled datasets of 7890 CICY 3-folds with all Hodge numbers [Green et al. (1987)]

Supervision and Function Approximation

\[ \begin{matrix} \mathcal{R}\colon & \mathbb{N}^{m \times k} & \longrightarrow & \mathbb{N}^2 \\[0.5em] & X & \mapsto & (h^{1,1},\, h^{2,1}) \\[0.5em] & \text{(conf. matrix)} & & \text{(Hodge no.)} \end{matrix} \]

Supervised Learning

replace \(\mathcal{R}( X )\) with \(\mathcal{R}( X;\, W )\) (\(W\) weights)
feed the algorithms \(X\) and \(h^{p,q}\) (true values)
\[ W = \arg\min\limits_{w}\; \mathcal{L}(h^{p,q},\, \mathcal{R}( X;\, w )) \]
follow gradient descent of \(\mathcal{L}\) \(\rightarrow\) tune \(W\)

Neural Networks as Function Approximators

Fully Connected Networks

older in design [Rosenblatt (1958)]
analogy neuron - “perceptron”
rose to fame in the 70s/80s
simple matrix multiplications \(+\) non linearity \[ a^{\{l+1\}} = \phi\left( w^{\{l\}} a^{\{l\}} + b^{\{l\}} \mathbb{I} \right) \in \mathbb{R}^p \] \[ \phi(z) = \mathrm{ReLU}(z) = \max(0, z) \]

Convolutional Neural Networks

newer in conception [LeCun et al. (1989)]
based on sliding windows (aka convolutions) \[ a^{\{l+1\}} = \phi\left( w^{\{l\}} * a^{\{l\}} + b^{\{l\}} \mathbb{I} \right) \in \mathbb{R}^{p \times q} \]
less parameters to isolate features

Inception Neural Networks

inspired by Google [Szegedy et al. (2014)]
created for computer vision
\(2 \times 10^5\) parameters (vs \(\ge 2 \times 10^6\))

concurrent kernels \(\Rightarrow\) shared parameters
retains “spatial awareness”
improved generalisation ability

Results

[Erbin, RF (2020)]

Comparison

[Erbin, RF (2020)]

Conclusion

[Machine learning is]

the field of study that gives computers the ability to learn without being explicitly programmed.

A. Samuel (1959)

deep learning can be a reliable predictive method
it can be used as source of inspiration for inference and generalisation
CNNs have a lot of unexpressed potential in physics (first time?)
the approach intersects mathematics, physics and computer science

What Lies Ahead?

improve \(h^{2,1}\) and “un-blackbox” the model (SHAP, filter analysis, etc.)
exploration for CICY 4-folds and representation learning (DGAN, VAE, RL, etc.)
study symmetries (GNNs, transformers architectures, etc.) and explore the string landscape

Thank You